Z-module homomorphism

Let (M a -module); let N be the Z-submodule of M generated by (where ). Let be a Z-module homomorphism such that g(N)={0}.
Use elements in M of the form and ( , in Z) to show that g=0. Hint: If , use that, given , there exist y in N, z in M such that to deduce that g(x)=0.

What confuses me is if g(N)=0, since an element x of M can be written as why then does it not follow that, since for each n, for any x in M? After that the purpose of the various k's is also a mystery to me.